state the values of x for which the following identiy is true
log(x+2) + log(x+3) = log(x^2+5x+6)
a) x> -2
b) x> -3
Why is the answer a and not b?
If it was b, then it would include -2
For the identity to be true, the values inside the log functions have to be greater than zero.
Try sketching the function x^2+5x+6 in the x-y plane, you will see that the value of the function is greater than zero when x>-2 and x<-3.
Now repeat the same for x+2 and x+3.
Would (a) or (b) satisfy all three conditions?
Ohh I see it now, thanks a bunch :P
To determine the values of x for which the given identity is true, we can use the properties of logarithms.
In this case, we have the logarithmic identity:
log(a) + log(b) = log(a * b)
Applying this identity to the given equation, we can simplify it as follows:
log(x+2) + log(x+3) = log(x^2+5x+6)
log((x+2)(x+3)) = log(x^2+5x+6)
Since the logarithms are equal, we can equate the arguments:
(x+2)(x+3) = x^2+5x+6
Expanding and simplifying this equation gives:
x^2 + 2x + 3x + 6 = x^2 + 5x + 6
x^2 - x^2 + 2x + 3x - 5x = 6 - 6
0x = 0
We get 0=0, which is true for all values of x.
Therefore, the given identity is true for all real values of x.
Now let's analyze the answer options:
a) x > -2: This option states that x should be greater than -2. Since the identity is true for all values of x, including values less than -2, option a) is correct.
b) x > -3: This option states that x should be greater than -3. Although it is true that if x > -3, it will also satisfy the identity, it is not the only condition for the identity to be true. As mentioned earlier, any real value of x satisfies the equation. Therefore, option b) is not the correct answer.
To summarize, the answer is option a) because the given identity is true for all values of x, not just for x greater than -3.